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Question Number 19609 by ajfour last updated on 13/Aug/17

Answered by ajfour last updated on 13/Aug/17

$$\mathrm{Equation}\mathrm{of}{\mathrm{L}}_1:$$$${\overline{\mathrm{z}}}_1\mathrm{z}+{\mathrm{z}}_1\overline{\mathrm{z}}-2\mid {\mathrm{z}}_1{\mid }^2=0$$$$\mathrm{Equation}\mathrm{of}{\mathrm{L}}_2:$$$${\overline{\mathrm{z}}}_2\mathrm{z}+{\mathrm{z}}_2\overline{\mathrm{z}}-2\mid {\mathrm{z}}_2{\mid }^2=0$$$$\mathrm{and}{\mathrm{z}}_0={\mathrm{z}}_1+{\mathrm{z}}_2$$$$\mathrm{As}{\mathrm{z}}_0\mathrm{lies}\mathrm{on}\mathrm{both}\mathrm{lines},$$$${\overline{\mathrm{z}}}_1({\mathrm{z}}_1+{\mathrm{z}}_2)+{\mathrm{z}}_1({\overline{\mathrm{z}}}_1+{\overline{\mathrm{z}}}_2)-2\mid {\mathrm{z}}_1{\mid }^2=0$$$$\Rightarrow \mid {\mathrm{z}}_1{\mid }^2+{\overline{\mathrm{z}}}_1{\mathrm{z}}_2+\mid {\mathrm{z}}_1{\mid }^2+{\mathrm{z}}_1{\overline{\mathrm{z}}}_2-2\mid {\mathrm{z}}_1{\mid }^2=0$$$$\mathrm{or}{\mathrm{z}}_1{\overline{\mathrm{z}}}_2+{\overline{\mathrm{z}}}_1{\mathrm{z}}_2=0.$$

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